Age 16 to 18 Challenge Level:
When you meet a class of equations like this it would be good to
find a general formula involving $a$ and $b$ (like the formula for
solutions of quadratic equations).
It often helps towards a general solution to try special cases as
suggested in this problem and exact solutions can be found in
special cases. More generally however there is no closed formula
for solutions and you will need to find approximate solutions by
This problem involves the use of logs, solving a quadratic equation
and using a numerical method such as interval halving or the Newton
Raphson method. To speed up the process of solving this equation
for a particular pair of values of $a$ and $b$ you can use a
programmable calculator or a spreadsheet.